Combinatorial Game Theory
who wins, and by how much
Concept

Witness — where it appears

A particular position exhibited to settle a claim, as against a count that supports one. A witness refutes outright and cannot be qualified by a wider sweep, which is why every negative result here carries one.

Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.

What the two outcome classes of the parts settle. For each pair of outcome classes, the set of outcomes the sums actually took. A cell with one letter is a pair of classes that decided the answer; a shaded cell with several is a pair that did not. Both conventions have ambiguous cells — the difference is that normal play repairs them with values and misère play has nothing to repair them with.

Two misère outcomes are not enough

Knowing who wins each part does not say who wins the sum. Over 676 sums built from a pool of twenty-six positions, nine of the sixteen pairs of outcome classes settle the answer under normal play and not one of the sixteen settles it under misère — and the nine that work are theorems about a value being zero, which is exactly the thing misère play does not have.

limits · Misère play What a wider pool rescues. The misère outcome table built four times over, on pools of 10, 22, 100, 113 positions. A cell holds the set of outcomes that sums of its row class and column class actually took. Fifteen of the sixteen cells are short of all four outcomes on the smallest pool and none is on the largest, so every near-miss in the original table was a statement about the pool rather than about misère play.

What a wider pool rescues

The misère outcome table has sixteen cells, and over a pool of ten positions fifteen of them hold fewer than four outcomes — which looks like structure and might be a shortage of positions. Thirteen values further on there is nothing left: every pair of outcome classes takes every outcome, so the near-misses were the pool, and the prediction the rung below made was right.

limits · Misère play When the regions add. Every board in the independence census — 117 positions whose empty points fall into two or more regions — tested against a stated criterion and against the guess it replaces. The criterion is that no stone group has liberties in two different regions, which makes a move in one region unable to change what is legal in another. It holds on 9 boards and the regions add on every one of them.

When the regions add

The rung below described the NoGo boards whose regions add as the ones with symmetric walls, and said the description was a guess made from six examples. It is wrong: fourteen symmetric boards do not add and sixteen that add are not symmetric. What replaces it is a criterion about liberties — sound on all 117 boards, provable in a line, and complete on only nine of the twenty-four.

positions · Nogo 6 turns in strict alternation, priced. One quantifier prefix taken apart into its blocks, with the size of a winning strategy computed a term at a time. A choice made after k of the opponent's turns has to be written down once for each of the 2^k lines the opponent can produce, so the total depends on where the opponent's turns sit and not merely on how many there are.

Twelve turns, and three different prices

The earlier essay prices a universal quantifier at a doubling and leaves it there. Twelve turns with six of them the opponent's cost 6, 63 or 384 decisions to write down, depending on nothing but the order the turns come in — and the cheap arrangements are cheap for only one of the two players. What a claim costs is the number of times the choosing changes hands.

complexity · Alternation A win is proved by one move and a loss by all of them. The smallest proof of each position's verdict, averaged by verdict. At a node the mover wins the proof takes the cheapest single option; at a node the mover loses it has to answer every option, which is the existential and universal quantifiers of the prefix showing up as two different objects.

Proving a loss means answering everything

A win is established by one move and a loss by every move, so the two verdicts are certified by objects of different shapes. Measured over every position of four games, a loss costs between 1.07 and 2.31 times a win — a small constant, never an exponential. The obvious explanation is the branching and it is wrong: Nim answers six options at a losing turn and pays 2.18, not six.

complexity · Alternation

Named alongside it

The objects these essays reach for when they reach for this one.

Exhaustive searchAdditivityDisjunctive sumNormal playOutcome classAlternationCertificateComplexityCounterexampleMisère playMisère quotientNegation

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